use quarto, not Pluto to render pages

CwJ/integrals/mean_value_theorem.jmd

@@ -98,10 +98,12 @@ Though not continuous, $f(x)$ is integrable as it contains only jumps. The integ
 What is the average value of the function $e^{-x}$ between $0$ and $\log(2)$?
 
 ```math
-\text{average} = \frac{1}{\log(2) - 0} \int_0^{\log(2)} e^{-x} dx
-= \frac{1}{\log(2)} (-e^{-x}) \big|_0^{\log(2)}
-= -\frac{1}{\log(2)} (\frac{1}{2} - 1)
-= \frac{1}{2\log(2)}.
+\begin{align*}
+\text{average} = \frac{1}{\log(2) - 0} \int_0^{\log(2)} e^{-x} dx\\
+&= \frac{1}{\log(2)} (-e^{-x}) \big|_0^{\log(2)}\\
+&= -\frac{1}{\log(2)} (\frac{1}{2} - 1)\\
+&= \frac{1}{2\log(2)}.
+\end{align*}
 ```
 
 Visualizing, we have
@@ -214,8 +216,8 @@ choices = [
 "``\\int_0^t (v(0) + v(u))/2 du = v(0)/2\\cdot t + x(u)/2\\ \\big|_0^t``",
 "``(v(0) + v(t))/2 \\cdot \\int_0^t du = (v(0) + v(t))/2 \\cdot t``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 
@@ -278,8 +280,8 @@ value of $g(x) = \lvert x \rvert$ over the interval $[0,1]$?
 choices = [
 L"That of $f(x) = x^{10}$.",
 L"That of $g(x) = \lvert x \rvert$."]
-ans = 2
-radioq(choices, ans)
+answ = 2
+radioq(choices, answ)
 ```
 
 
@@ -297,8 +299,8 @@ choices = [
 ]
 n1, _ = quadgk(x -> x^2 *(1-x)^3, 0, 1)
 n2, _ = quadgk(x -> x^3 *(1-x)^4, 0, 1)
-ans = 1 + (n1 < n2)
-radioq(choices, ans)
+answ = 1 + (n1 < n2)
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -324,8 +326,8 @@ choices = [
 L"Because the mean value theorem says this is $f(c) (x-a)$ for some $c$ and both terms are positive by the assumptions",
 "Because the definite integral is only defined for positive area, so it is always positive"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 * Explain why $F(x)$ is increasing.
@@ -336,8 +338,8 @@ L"By the extreme value theorem, $F(x)$ must reach its maximum, hence it must inc
 L"By the intermediate value theorem, as $F(x) > 0$, it must be true that $F(x)$ is increasing",
 L"By the fundamental theorem of calculus, part I, $F'(x) = f(x) > 0$, hence $F(x)$ is increasing"
 ]
-ans = 3
-radioq(choices, ans)
+answ = 3
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -354,6 +356,6 @@ choices = [
 L"The average of $f$",
 L"The exponential of the average of $\log(f)$"
 ]
-ans = val1 > val2 ? 1 : 2
-radioq(choices, ans)
+answ = val1 > val2 ? 1 : 2
+radioq(choices, answ)
 ```